Theorem of Calculus

Proposition (First Fundamental Theorem of Calculus) If is a continuous function on the interval and is any anitderivative of throughout the interval then

    The hypothesis of the first fundamental theorem should not be overlooked; for example, if we apply the first fundamental theorem to evaluate

  

then we obtain something which is obviously not true since is never negative. Since is not continuous on the first fundamental theorem should not be used.

Example (First Fundamental Theorem of Calculus) Given , evaluate



    Solution. Since using the fundamental theorem of calculus,  
    

The previous examples demonstrates that some nice integral equations can be setup by using trigonometric identities, for example, since we can evaluate

Example (First Fundamental Theorem of Calculus) Evaluate



    Solution. Since on and is continuous on we can use the Fundamental Theorem of Calculus,  
    

    

Example (First Fundamental Theorem of Calculus) Evaluate



    Solution. Note that
    


using the Fundamental Theorem of Calculus and the Subdivision Rule,   

Example (First Fundamental Theorem of Calculus) Evaluate



    Solution. Because is a continuous function on , we can use the Fundamental Theorem of Calculus,
    

Example (First Fundamental Theorem of Calculus) Evaluate



    Solution. Because is a continuous function, we can use the Fundamental Theorem of Calculus,

Example (First Fundamental Theorem of Calculus) Evaluate

   where   

    Solution. We break the integral up into two pieces. On each piece we have a continuous function on a closed bounded interval and so we can apply the Fundamental Theorem of Calculus,  
    

Proposition (Integration by Substitution) If is a continuous function of and is a differentiable function of , then

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

    Solution. Let and so Then

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

    Solution. Let then and so

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

    Solution. Using the fundamental theorem of calculus,





For the second integral we let and so Therefore,








    

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

    Solution. First we rewrite,
    


Now let and then and so












    

Example (Integration by Substitution with a Definite Integral) The slope at each point on the graph of is given by What is if the graph passes through the point

    Solution. We are given and so Let then









Now allows us to determine by way of and therefore

Example (Integration by Substitution with a Definite Integral) Evaluate

    Solution. First factor by grouping to obtain,









Then let and so and then







Then let and so and then

Proposition (Second Fundamental Theorem of Calculus) Let be a continuous function on the interval and let be the function defined by the rule



Then is the anitderivative of on that is,



on

Example (Second Fundamental Theorem of Calculus) Differentiate the function



    Solution.  Since is continuous on the interval where is any real number greater than 6, we see that the second fundamental theorem applies to the interval and so,

Example (Second Fundamental Theorem of Calculus) Differentiate the function



    Solution.  Since


    
and is continuous on the interval where is any real number greater than 12, we see that the second fundamental theorem applies to the interval and so,

Example (Second Fundamental Theorem of Calculus) Find using the second fundamental theorem of calculus.

    Solution. Let and we will use the second fundamental theorem of calculus,









We can check  



Thus,  

Example (Second Fundamental Theorem of Calculus) Find using the second fundamental theorem of calculus.

    Solution. Since is continuous when we can apply the Second Fundamental Theorem,

Example (Second Fundamental Theorem of Calculus) Find an equation for the tangent line to the curve at the point where and

    Solution. Since is continuous on we can apply the Second Fundamental Theorem of Calculus,

  

Thus the slope of the tangent line to the curve at the point where is So an equation of the tangent line is which simplifies to or standard form

Example (Second Fundamental Theorem of Calculus) Suppose Find   

    Solution. Assume Since is continuous when we can apply the Second Fundamental Theorem, with
    


and since the derivative is a linear function,



We apply the Second Fundamental Theorem to each,

Example (Second Fundamental Theorem of Calculus) Let ,  where is a function whose graph is shown. Estimate and Find the largest open interval on which is increasing. Find the largest open interval on which is decreasing. Identify any extrema of



    Solution.  First,
    










The largest open interval where is increasing is and the largest open interval on which is decreasing is Since is continuous on the second fundamental theorem yields defined on Therefore, a maximum occurs when which is at

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Theorem Of Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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